S -Almost Automorphic Solutions for Impulsive Evolution Equations on Time Scales in Shift Operators

: In this paper, based on the concept of complete-closed time scales attached with shift direction under non-translational shifts (or S -CCTS for short), as a ﬁrst attempt, we develop the concepts of S -equipotentially almost automorphic sequences, discontinuous S-almost automorphic functions and weighted piecewise pseudo S -almost automorphic functions. More precisely, some novel results about their basic properties and some related theorems are obtained. Then, we apply the introduced new concepts to investigate the existence of weighted piecewise pseudo S -almost automorphic mild solutions for the impulsive evolution equations on irregular hybrid domains. The obtained results are valid for q -difference partial dynamic equations and can also be extended to other dynamic equations on more general time scales. Finally, some heat dynamic equations on various hybrid domains are provided as applications to illustrate the obtained theory.


Introduction
Almost automorphic functions, which are more general than the almost periodic functions, were introduced by Bochner (see [1][2][3]) in relation to some aspects of differential geometry. Almost automorphic solutions in the context of differential equations have been studied by several researchers. For instance, pseudo and weighted pseudo almost automorphic mild solutions to (fractional) evolution equations were investigated by Chang et al. [4][5][6], Ding et al. [7,8], Diagana [9,10]. Subsequently some interesting properties of the space of weighted pseudo almost automorphic functions like the completeness and the composition theorem were reported in [11,12] by N'Guérékata which have many applications in the context of differential equations. For more details about this topic we refer to the recent books (see [10,11]), where the authors gave important overviews about the theory of almost automorphic functions and their applications to differential equations.
Since time-scale calculus was proposed by Hilger [13], Bohner and Guseinov have extensively developed this theory on the aspect of integral and dynamic equations (see [14,15]). To study the approximation properties of time scales, some new concepts such as almost periodic time scales and changing-periodic time scales were proposed and studied by Agarwal et al. (see [16,17]). In addition to these fundamental results, there have been many works on different types of dynamic equations on time scales. For example, the concept of variable time scales was introduced and a novel idea of the mutual transformation between impulsive dynamic equations and dynamic equations on variable time scales was initiated by Akhmet et al. [18][19][20]. In the literature [21], Bohner et al. established an SIR model on the general time scales and derived its exact solution. In [22], the existing ideas of the univariate case of the time-scale calculus was generalized to the bivariate case and applied to partial dynamic equations. In the stability analysis, Martynyuk and Stamova investigated the sets of dynamic equations and hybrid dynamic systems on time scales (see [23,24]). In [25,26], two types of new high order derivations were introduced and the existence of solutions for the type high order fractional integro-differential equations was studied by Aydogan and Baleanu et al., and these types of fractional corresponding derivatives were generalized to time scales by Mozyrska, Ortigueira and Torres et al. (see [27,28]). In the field of studying functions and applications, it is a hot topic to study the almost automorphic and almost periodic functions and applications to dynamic equations based on time scales. For example, Hong investigated the almost periodic set-valued functions and almost periodic set dynamic equations on time scales (see [29]). On almost automorphic functions and its related problems, Kéré, Mophou, N'Guérékata et al. investigated (n-order ) almost automorphic and asymptotically almost automorphic functions of n-order, some basic results were obtained and applied to abstract dynamic equations (see [30][31][32]). In 2020, based on the concepts the authors introduced on translation time scales, Wang et al. established a theory of closedness of translation time scales and their applications to evolution equations and dynamical models (see the monograph [33]). In addition, a new concept of periodic time scales and the notion of shift operators of time scales were proposed and studied under the background of studying periodic functions (see Adıvar et al. [34,35]). It is easy to observe that periodic time scales under translations have a nice closedness and their graininess function µ is bounded.
However, some classical and important time scales are irregular and they have the unbounded graininess function µ. For example, T = q N 0 := {q t : t ∈ N 0 for q > 1} ∪ {0}, where N 0 is the set of natural numbers or T = q Z := q Z ∪ {0} or quantum-like time scale T = (−q) Z (which has applications in quantum theory) and other types of time scales like T = N 2 and T = T n the space of the harmonic numbers (it is of interest to study almost automorphic dynamic behavior of solutions for q-difference-like dynamic equations among others, see Wang et al. [36][37][38]). It is impossible to introduce almost automorphic functions on such a type of time scale since the translation approximation of functions will never be reached for the reason that the graininess function µ is a strictly increasing function for time scales. In addition, many natural phenomena must be modeled as a process in which continuous evolution is usually interrupted by an event (impulses, catastrophe, etc., see Stamova [39,40] and Wang et al. [41][42][43]), which motivates us to investigate general evolution equations with impulses on irregular hybrid domains.
In the present paper, for the first time, we study the existence of weighted piecewise pseudo S-almost automorphic mild solutions for the impulsive evolution dynamic equations where A ∈ PC rd T, B(X) is a bounded linear operator in the Banach space X and f ∈ PC rd (T × X, X), x σ = x σ(t) . f , I k , t k satisfy suitable conditions that will be established later and T is a complete-closed time scale attached with shift direction under non-translational shifts (S-CCTS).
In addition, the notations x(t + k ) and x(t − k ) represent the right-hand and the left-hand side limits of x(·) at t k , respectively. In addition, some Lemmas are obtained and the exponential stability of weighted piecewise pseudo S-almost automorphic mild solutions is also studied. Finally, we apply these obtained results to study a class of ∆-partial differential equations on S-CCTS. The obtained results in this paper are feasible and effective on q-difference partial dynamic equations and more.
For instance, in (1), by using the shift operators δ ± in Section 2, (ii) if we let T = {q n : q > 1, n ∈ Z} = q Z and t k = q k 3 , k ∈ Z, then (1) turns into (iii) if we let T = hZ, h > 0 and t k = hk 3 , k ∈ Z, then (1) turns into where q t = q 2 if t > 0 and q t = 1/q 2 if t < 0, it is a classical q-dynamic system on quantum-like hybrid domains.
We provide four types of impulsive evolution dynamic equations in the above, in fact, (1) will turn into other different types of dynamic equations on different types of complete-closed time scales attached with shift direction under translational or non-translational shifts.
The highlights of the paper can be summarized as follows

•
We introduce the concept of S-equipotentially almost automorphic sequences under S-CCTS.

•
We establish a theory of discontinuous S-almost automorphic functions and weighted piecewise pseudo S-almost automorphic functions. Some new results about their basic properties and some related theorems are obtained. • The existence of weighted piecewise pseudo S-almost automorphic mild solutions for the impulsive evolution equations on irregular hybrid domains is studied. • The obtained results in this paper are effective for q-difference heat equations and other dynamic equations on more general hybrid domains.

S-Equipotentially Almost Automorphic Sequence Under S-CCTS
In this section, we will introduce some knowledge of complete-closed time scales under non-translational shifts (or S-CCTS for short) and then define S-equipotentially almost automorphic sequence and study its properties. For more details about time-scale calculus and S-CCTS, one may refer to the book [14,17].
For convenience, we introduce the notations. Let T * be the largest open subset of T, i.e., T * = T.

Remark 2.
Note that if T is a periodic time scales under translations and Π ± ⊆ T * , then the shift operators If T is a bi-direction S-CCTS and t 0 is the initial point, then for any s ∈ Π ± , we define a function A : Π ± → Π ± , which will be used later. Note that A(s) > t 0 and A(s) ≥ s.

Example 3.
The time scale q Z = q n : n ∈ Z and q = 3 √ 3 ∪ {0} is periodic with period τ under the shift operator δ ± .
Step 1. Periodic function construction. The piecewise periodic function defined by Step 2. Almost periodic function construction. Through Step 1, we can obtain an almost periodic piecewise function ln qt ln q on q Z , where f 1 , f 2 are periodic piecewise functions on q Z , respectively. Note that the periods of f 1 and f 2 are completely different.
Step 3. Almost automorphic function construction. According to the above, letF which is an almost automorphic function on q Z .
Next, based on Definition 1, we will introduce the concept of S-equipotentially almost automorphic sequence and study its properties.
Remark 4. If T is a periodic time scale under translations, then one can obtain the classical derivative sequence of {t k } satisfying t j k = t k+j − t k by letting δ ± (s, t) = t ± s. Particularly, if T = R, one can obtain the derivative sequence of {t k } from [40] (pp. 191-194) immediately.

Lemma 1 ([37]
). If T * be the largest subset of T and including a fixed number t 0 ∈ T * such that there exist operators δ ± : T * → T * , then Based on the S-derivative sequence and its properties, we will propose the following definition.
Definition 3. Let T be a S-CCTS under shifts δ ± and t j k = δ − (t k , t k+j ), k, j ∈ Z. The sequence {t j k }, k, j ∈ Z is said to be S-equipotentially almost automorphic if for any sequence {s n } ⊂ Z, there exists a subsequence {s n } such that Remark 5. In Definition 3, note that T is a closed subset of R, thus, for each k ∈ Z, one has γ k ∈ T.

S-Almost Automorphic Functions and Weighted Pseudo S-Almost Automorphic Functions
Let X be a Banach space endowed with the norm · . B(X, Y) denotes the Banach space of all bounded linear operators from X to Y. This is simply denoted as B(X) when X = Y. BC(T, X) is the space of bounded continuous function from T to X equipped with the supremum norm defined by u ∞ = sup t∈T u(t) .
In the following, we will give the definition of rd-piecewise continuous functions on time scales.

Definition 4.
We say ϕ : T → X is rd-piecewise continuous with respect to a sequence {t k } ⊂ T which satisfy

called intervals of continuity of the function ϕ(t).
For convenience, PC rd (T, X) denotes the set of all rd-piecewise continuous functions with respect let Ω be a subset of X and Now, let T, P ∈ B and let s(T ∪ P) : B → B be a map such that the set s(T ∪ P) forms a strictly increasing sequence. For For convenience, consider the metric space BPC rd (T, X) × B with the metric Proof. For any given Cauchy sequence φ n = ϕ n (t), T n ⊂ BPC rd (T, X) × B, we can obtain that the sequences {ϕ n (t)} and {T n } be the Cauchy sequences in the metric space (BPC rd (T, X), · ∞ ) and (B, d) respectively. Hence, for any ε > 0, there exists some n 0 > 0 such that n, m > n 0 implies d(T n , T m ) < ε, which yields that |t Moreover, for n > n 0 , we can obtain We claim that ϕ(t) is also bounded on T\F ε s(T n ∪ T) . In fact, there exists some n 0 > 0 such that To complete the proof, it is sufficient to show ϕ n → ϕ in norm on T\F ε s(T n ∪ T) , i.e., ϕ − ϕ n → 0 as n → ∞. According to (5), there exists some n 0 > 0 such that n, m > n 0 implies for all t ∈ T\F ε s(T n ∪ T) , so ϕ − ϕ n ∞ ≤ ε. This completes the proof.
Definition 6. Let T be a bi-direction S-CCTS. A function ϕ ∈ BPC rd (T, X) is said to be rd-piecewise S-almost automorphic if the following conditions are fulfilled: for each t ∈ T. Denote by AA S (T, X) the set of all such functions.
(iii) A bounded function f ∈ BPC rd (T × X, X) with respect to a sequence T = {t k } is said to be piecewise S-almost automorphic if f (t, x) is piecewise S-automorphic in t ∈ T uniformly for x ∈ B, where B is any bounded subset of X. Denote by AA S (T × X, X) the set of all such functions.
Similarly, we can also introduce the concept of piecewise S-almost automorphic functions that belong to PC rd (T, X).
Let U be the set of all functions ρ : T → (0, ∞) which are positive and locally ∆-integrable over T. For a given r ∈ (0, ∞) Π ± and ∀t 0 ∈ T * , set for each ρ ∈ U.

Remark 7.
Particularly, if we let T = {q n : n ∈ Z, q > 1} ∪ {0} and t 0 = q, then (6) will turn into the integral on the quantum time scale: Moreover, let T = {± √ n : n ∈ N} and t 0 = 1, then the integral is Similarly, we define We are now ready to introduce the sets WPAA S (T, ρ) and WPAA S (T × X, ρ) of weighted pseudo S-almost automorphic functions: and φ ∈ WPAA S 0 (T × X, ρ) .

Lemma 3. Let T be bi-direction S-CCTS under shifts
Proof.
By Definition 6, the following two Lemmas are obvious.

Lemma 7.
Let T be bi-direction S-CCTS under shifts δ ± . The decomposition of a weighted piecewise pseudo S-almost automorphic function according to AA S ⊕ WPAA S 0 is unique for any ρ ∈ U B .
Similarly, the same step can be applied for lim n→∞φ * −τ n =φ pointwise on T, thus, we can obtain the desired result. This completes the proof.

Theorem 2. Let T be bi-direction S-CCTS under shifts
is a closed subspace of BPC rd (T × Ω, X). Therefore, WPAA S 0 (T × X, ρ) is itself a Banach space. Then by Lemmas 7 and 8, we have WPAA S (T × X, ρ), · ∞ is a Banach space. The proof is completed.
By Lemmas 3 and 9, one can get the following theorem immediately without proof.
Next, we will show the following two Lemmas, which are useful in the proof of our results. Lemma 10. Let T be bi-direction S-CCTS under shifts δ ± . If ϕ ∈ PC rd (T, X) is an S-almost automorphic function with respect to the sequence T and {t k } ⊂ T is S-equipotentially almost automorphic satisfying inf k∈Z t q k = θ > t 0 , q ∈ Z, where t 0 is an initial point, then ϕ(t k ) is an S-almost automorphic sequence in X.
Proof. Let t j k = δ − (t k , t k+j ), k, j ∈ Z. Obviously, from the definition of Π ± , it is easy to know that t j k ∈ Π ± . Since ϕ ∈ PC rd (T, X) is an S-almost automorphic function and {t k } ⊂ T is S-equipotentially almost automorphic, from Definitions 3 and 6, for any sequence {s n } ⊂ Z, there exists a subsequence {s n } such that Hence, {ϕ(t k )} is an S-almost automorphic sequence in X. This completes the proof.
By Lemma 11, we can straightly get the following theorem: Theorem 5. Let T be bi-direction S-CCTS under shifts δ ± and δ + be ∆-differentiable to its second argument with δ ∆ + (s, ·) <δ ∆ + (s ∈ Π − ), whereδ ∆ + is a positive number. A necessary and sufficient condition for a bounded sequence {a n } to be in WPAA S (Z, ρ) is that there exists a uniformly continuous function f ∈ WPAA S (T, ρ) such that f δ n r(n) (t 0 ) = a n , t 0 ∈ T, n ∈ Z, ρ ∈ U B . Theorem 6. Let T be bi-direction S-CCTS under shifts δ ± and δ + be ∆-differentiable to its second argument with δ ∆ + (s, ·) <δ ∆ + , whereδ ∆ + is a positive number. Assume that ρ ∈ U B and the sequence of vector-valued functions {I k } k∈Z is weighted pseudo S-almost automorphic, i.e., for any x ∈ Ω, {I k (x), k ∈ Z} is weighted pseudo S-almost automorphic sequence.
It follows from Lemma 10 that the sequence φ 1 (t k ) is S-almost automorphic. To show h(t k ) is weighted pseudo S-almost automorphic, we need to show that φ 2 (t k ) ∈ WPAA S 0 (Z, ρ). By the assumption, h, φ 1 ∈ UPC(T, X), so is φ 2 . Let 0 < ε < 1, there exists δ * > 0 such that for t ∈ (t k , t k + δ * ) T , k ∈ Z, we have Without loss of generality, let t n ≥ t 0 , t −n < t 0 , n ∈ Z + , there exists r n , r −n ∈ (t 0 , ∞) Π ± such that δ + (r n , t 0 ) = t n , δ − (r −n , t 0 ) = t −n . Let r n = max{r n , r −n } ∈ Π ± . Therefore, repeating the proof of Lemma 11, we can obtain h(t k ) is weighted pseudo S-almost automorphic. Now, we show I k h(t k ) is weighted pseudo S-almost automorphic. Let Since I k , h(t k ) are two weighted pseudo S-almost automorphic, by Lemma 11 and Theorem 5, we know that I ∈ WPAA S (T × Ω, ρ), Φ 0 ∈ WPAA S (T, ρ). For every t ∈ T, there exists a number k ∈ Z such that t − δ k r(k) (t 0 ) ≤ r, Since {I k (x) : k ∈ Z, x ∈ K} is bounded for every bounded set K ⊆ Ω, {I(t, x) : t ∈ T, x ∈ K} is bounded for every bounded set K ⊆ Ω. For every x 1 , x 2 ∈ Ω, we have Noting that I k (x) is uniformly continuous in x ∈ Ω uniformly in k ∈ Z, we then get that I(t, x) is uniformly continuous in x ∈ Ω for t ∈ T. Then by Theorem 4, I ·, Φ 0 (·) ∈ WPAA S (T, X). Again, using Lemma 11 and Theorem 5, we have that I δ k r(k) (t 0 ), Φ 0 (δ k r(k) (t 0 )) is a weighted pseudo S-almost autmorphic sequence, that is, I k h(t k ) is weighted pseudo S-almost automorphic. This completes the proof.
From Theorem 6, one can easily get the following corollary: Corollary 2. Let T be bi-direction S-CCTS under shifts δ ± and δ + be ∆-differentiable to its second argument with δ ∆ + (s, ·) <δ ∆ + , whereδ ∆ + is a positive number. Assume the sequence of vector-valued functions {I k } k∈Z is weighted pseudo S-almost automorphic, ρ ∈ U B , if there is a number L > 0 such that for all x, y ∈ Ω, k ∈ Z and h ∈ WPAA S (T, ρ) ∩ UPC(T, ρ) such that h(T) ⊂ Ω, then I k h(t k ) is a weighted pseudo S-almost automorphic sequence.

Weighted Piecewise Pseudo S-Almost Automorphic Mild Solutions to the Impulsive ∆-Evolution Equations
In this section, we investigate the existence and exponential stability of a piecewise weighted pseudo S-almost automorphic mild solution to Equation (1). For this, we will provide a Lemma that will be used in our main results.

Remark 8.
It is easy to observe that if µ(t) is bounded, then there exists a sufficiently small constant β 1,α > 0 and a sufficiently large constant β 2,α > 0 such that (12) is valid. Therefore, Lemma 12 holds when T is an almost periodic time scale from [33].
Consider the impulsive linear ∆-evolution equation where A : T → B(X) is a linear operator in the Banach space X. We denote by B(X, Y) the Banach space of all bounded linear operators from X to Y. This is simply denoted as B(X) when X = Y. (1) T(s, s) = Id, where Id denotes the identity operator in X; (2) T(t, s)T(s, r) = T(t, r); (3) the mapping (t, s) → T(t, s)x is continuous for any fixed x ∈ X.

Definition 9.
A function x : T → X is called a mild solution of Equation (1) if for any t ∈ T, t > c, c = t k , k ∈ Z, one has In the following, consider (1) with the following assumptions: (H 1 ) Let T be bi-direction S-CCTS under shifts δ ± and δ + be ∆-differentiable to its second argument with δ ∆ + (s, ·) <δ ∆ + (s ∈ Π − ), whereδ ∆ + is a positive number.
(H 2 ) The family {A(t) : t ∈ T} of operators in X generates an S-exponentially stable evolution system {T(t, s) : t ≥ s}, i.e., there exist K 0 > 1 and ω > 0 such that e ω (t, s), t ≥ s, ω ∈ R + . (14) ( where ρ ∈ U B and f (t, ·) is uniformly continuous in each bounded subset of Ω uniformly in t ∈ T; I k is a weighted pseudo S-almost periodic sequence, I k (x) is uniformly continuous in x ∈ Ω uniformly in k ∈ Z, inf k∈Z t 1 k = δ − (t k , t k+1 ) = θ > max{t 0 , 0}, where t 0 is the initial point.

Remark 9.
In the assumption (H 2 ), let T = R, Equation (14) turns into Moreover, let the time scale T be the quantum time scale q Z = {q n : q > 1, n ∈ Z}, then To investigate the existence and uniqueness of a weighted piecewise pseudo S-almost automorphic mild solution to Equation (1), we need the following Lemma: Lemma 13. Assume v ∈ AA S (T, X), ω ∈ R + and (H 1 ) − (H 3 ) are satisfied. If u : T → X is defined by where v n (s) = v δ + (τ n , s) , n = 1, 2, . . ..
(A 2 ) f ∈ WPAA S (T × Ω, ρ), and f satisfies the Lipschitz condition with respect to the second argument, i.e., (A 3 ) I k is a weighted pseudo S-almost periodic sequence, and there exists a number L 2 > 0 such that for all x, y ∈ Ω, k ∈ Z.
Proof. Consider the nonlinear operator Γ given by By Theorem 7, we see that Γ maps WPAA S (T, ρ) into WPAA S (T, ρ).

Applications
In this section, three examples are demonstrated as the applications of our obtained results. Example 4. Let T 1 be S-CCTS under shifts δ ± and u : T 1 × T 2 → R, where the hybrid domain T 1 is the time scale in Example (1) and T 2 is a discrete time scale with a forward jump operator σ 2 and 0, π ∈ T 2 , i.e., T 1 = (−q) Z = (−q) n : q > 1, n ∈ Z ∪ {0}, where q = √ 3.

Conclusions and Open Problems
In this paper, we have introduced a concept of complete-closed time scales attached with a shift direction under non-translational shifts (S-CCTS). This is the first attempt to introduce and study the concepts of S-equipotentially almost automorphic sequence, discontinuous S-almost automorphic functions and weighted piecewise pseudo S-almost automorphic functions. Then, we apply the introduced concepts to investigate the existence of weighted piecewise pseudo S-almost automorphic mild solutions for a class of evolution impulsive equations on hybrid domains. Finally, we apply the obtained results to ∆-partial dynamic equations from which one can see the established results are feasible and effective for q-difference partial dynamic equations among others. It is obvious that these results are more general and comprehensive than previous literature.
On the other hand, by virtue of S-CCTS, the almost automorphic problems of q-difference partial dynamic equations can also be introduced and studied. In fact, from the construction of almost automorphy of functions in Section 3, one can also establish some new types of functions with almost automorphy. Moreover, by introducing the concepts of almost automorphic functions and ∆-almost automorphic functions and developing an appropriate approach, we will study almost automorphic problems of ∆-partial dynamic equations on irregular time scales, which will be the topic of our future work.
Based on our discussion, we introduce the following open problems: (1) For the highly hybrid time scales such as T = q Z ∪ {hZ} ∪ N 1 2 0 , how to provide an effective way to construct the shift operators? (2) How to establish a feasible method to study the almost automorphy of solutions to discontinuous dynamic systems on highly hybrid time scales?
Author Contributions: All authors contributed equally and significantly in this paper and typed, read, and approved the final manuscript. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors declares no conflict of interest.